Internet | Account Information | DNS server defines where your Fritzbox to send its DNS requests.
By default, your FB will tell its clients to use the FB for DNS, and thus, all your clients DNS requests will ultimately be forwarded to those DNS servers, minus the few queries the FB can itself, i.e. for local names like fritz.box.
Since you've configured your Fritzbox to instruct its clients to use your Pi-hole machine's IP for DNS, your FB won't receive any DNS requests from your home network, only using its upstream DNS servers for its own requests (like firmware updates) and for its guest network (if that's enabled).
If you enable Conditional Forwarding to your FB, your Pi-hole will query your FB for local names. Your FB will answer them if it knows them, but if it doesn't, it forward them upstream. If that's Pi-hole, Pi-hole would then conditionally forward them to your FB, closing the partial DNS loop.
As for "failed to send TCP(read_write) packet (Connection prematurely closed by remote server)" messages triggered by unbound, see TCP connection failed while receiving payload length from upstream - #2 by Bucking_Horn.